11). The ratio of length of each equal side and the third side of an isosceles triangle is 3:4. If the area of the triangle is \( \Large \Large 18\sqrt{5} \) sq units, the third side is

A). \( \Large 8\sqrt{2} \)units

B). 12 units

C). 16 units

D). \( \Large 5\sqrt{10} \)units

View Answer

Correct Answer: 12 units

Let sides of isosceles triangle are 3x,3x and 4x.

Then, half-perimeter (s) = \( \Large \frac{a+b+c}{2} \)

= \( \Large \frac{3x+3x+4x}{2} \)=5x

Given, area of isosceles triangle

= \( \Large 18\sqrt{5} \)sq units

\( \Large \sqrt{s(s-a)(s-b)(s-c)}=18\sqrt{5} \)

\( \Large \sqrt{5x(5x-3x)(5x-3x)(5x-4x)}=18\sqrt{5} \)

\( \Large \sqrt{5x\times 2x\times 2x\times x}=18\sqrt{5} \)

=> \( \Large 2\sqrt{5}x^{2}=18\sqrt{5} \)

=> \( \Large x^{2}=9 \)

=> x=3

Third side of isosceles triangle

=\( \Large 4\times 3 \)= 12units

12). Three sides of a triangular field are of length 15 m, 20 m and 25 m long, respectively. Find the cost of sowing seeds in the field at the rate of RS. 5 per Sq m.

A). 750

B). 150

C). 300

D). 600

View Answer

Correct Answer: 750

Since,\( \Large AC^{2}=AB^{2}+BC^{2} \)

=> \( \Large (25)^{2}=(15)^{2}+(20)^{2} \)

=> 625=225+400

=> 625 = 625

So, the triangular field is right angled at B

Area of the field = \( \Large \frac{1}{2}\times AB\times BC \)

= \( \Large \frac{1}{2}\times 15\times 20 \)

=\( \Large 150m^{2} \)

So, the cost of sowing seed is RS. 5 per sq m

Cost of sowing seed for

\( \Large 150m^{2}=150\times 5 \)

= RS. 750

13). A \( \Large \Large \triangle DEF \) is formed by joining the mid points of the sides of \( \Large \Large \triangle ABC\). Similarly, a \( \Large \Large \triangle PQR \) is formed by joining the mid-points of the sides of the \( \Large \Large \triangle DEF \). If the sides of the PQR are of lengths 1, 2 and 3 units, what is the perimeter of the \( \Large \Large \triangle ABC \)?